CTI, November 19, 2015

Transcription

1 Consider a large cube made from unit cubes 1 Suppose our cube is n n n Look at the cube from a corner so that you can see three faces How many unit cubes are in your line of vision? Build a table that shows how many cubes are visible from one corner n n 3 number visible How does the table continue? Make some guesses and then try to prove your answer Let s name the number we re looking for Let G(n) denote the number of cubes visible from a corner of the n n n cube Notice that the sequence of differences G(2) G(1) = 6; G(3) G(2) = 19 7 = 12; G(4) G(3) = = 18 has an interesting property The differences are all multiples of 6 When we explore such a sequence in which the sequence of successive differences is eventually constant, we can build a polynomial that produces the sequence Since the second order differences are constant, we propose that G(n) is a quadratic polynomial, G(n) = an 2 + bn + c We can solve this without great difficulty to get G(n) = 3n 2 3n + 1 But what do these coefficients have to do with the problem? One way to see this is to extend the chart by one more column that shows the cubes that are not visible n n 3 number not visible number visible Now you can see that we can count the number of visible cubes by counting the invisible ones first So, in general, G(n) = n 3 (n 1) 3 = n 3 (n 3 3n 2 + 3n 1) = 3n 2 3n + 1 This doesn t completely answer the question however What are the coefficients telling us? How does the algebra help us reason geometrically? The answer is below But before we leave this rich area, we have one more question to pursue If we paint the entire outside, how many of the n 3 unit cubes receive some paint? Can you 1 Unauthorized reproduction/photocopying prohibited by law c 1

2 write this as a polynomial in n in standard form What do the coefficients tell you? Problems with Cubes 1 The entire outside of an n n n cube built from unit cubes is painted Find the number of cubes with some painted faces as a function of n Note that for n = 3, the number is 27 1 = 26 n n 3 number with paint Suppose that every pair of interior faces are glued together so that each pair of faces requires one unit of glue How many square units of glue is needed? Examine this for n = 4, 5, and 6 3 What is the minimum amount of glue needed to hold together all n 3 cubes, for n = 2, 3, 4, 5 This brings up the issue of whether a cube is rigid if the surface is rigid We can try this problem under both assumptions, one where we simply need make the surface rigid and the other where every cube must get some glue Try these for n = 3 where only the surface of the cube need be rigid 4 What is the fewest cuts needed to separate a wooden cube into 27 unit cubes if you re allowed to move blocks of cubes about before cutting? What if the big cube is 4 4 4? 5 Suppose all six faces of an a b c, a b c block of unit cubes are painted and it turns out that exactly the same number of unit cubes have some paint as those that have no paint Find all triplets (a, b, c) for which this occurs Prove that there are no other solutions State and solve the planar analog of this problem 6 Suppose all six faces of an n n n are painted red Then one of the n 3 unit cubes is randomly selected and tossed like a die What is the probability that the face obtained is painted? Of course, your answer depends on n Try this for n = 1, 2, and 3 Then make a conjecture and prove your conjecture 2

3 7 Suppose all six faces of an a b c block of cubes are painted red Then one of the abc unit cubes is randomly selected and tossed like a die Is it possible that the probability of a red face showing up is exactly 2/7? This problem clearly generalizes the previous one Show that when a = b = c, we get from this problem the same result as we got in the one above 8 Let n = 3 Randomly and simultaneously select two unit cubes and toss them What is the probability that they both have painted faces showing? 9 Suppose two non-adjacent faces of the big cube are painted red and the other four faces painted black Let R denote the number of unit cubes with some red faces and B the number of unit cubes with some black faces Find an n for which B R = (2008 Mathcounts) A cube is built using a cube and a bunch of cubes How many cubes are needed? A (n + 2) (n + 2) (n + 2) cube is built using a n n n cube, a bunch of cubes and a few unit cubes How many unit cubes and cubes are needed? Your answer may depend on the oddness or evenness of n 11 A mouse eats away at a block of cheese that is made from 27 units of cheese 14 unit cubes are dark cheese and 13 unit cubes are a light colored cheese and they are arranged so that no two cubes of the same cheese have a common face In other words its colored like a checkerboard The mouse proceeds always to an adjacent cube (Adjacent means that the two small cubes share a face) Can the mouse eat in such a pattern so that she finishes her meal on the centermost cube? 12 A square can be partitioned into four squares in an obvious way A square can also be partitioned into seven squares and nine squares What is the largest integer N such that a square cannot be partitioned into N squares? 13 You have an unlimited supply of red(r) and blue(b) faces out of which to build cubes How many distinguishable cubes can you build? Next suppose you have three colors 3

4 14 Bob and Ann play the following game with 8 white unit cubes Ann wins if she can assemble a cube that has only white faces exposed But Bob gets to paint four of the 8 6 = 48 white faces black Who wins? (a) What is the fewest number of faces Bob can paint to deny Ann in the game? (b) What is the fewest number of faces Bob can paint to deny Ann in the game? (c) What is the fewest number of faces Bob can paint to deny Ann in the game? 15 Chameleon Cubes You re given 8 unpainted cubes Can you paint the faces with two colors, red and blue, so that when you re done, you can assemble both an all red cube and an all blue cube? Chameleon Cubes You re given 27 unpainted cubes Can you paint the faces with three colors, red, white, and blue, so that when you re done, you can assemble an all red cube, an all white cube and an all blue cube? Chameleon Cubes You re given 64 unpainted cubes Can you paint the faces with four colors, red, white, green and blue, so that when you re done, you can assemble cubes of all four colors? 18 Can you devise a planar version of the problem above? How about a one-dimensional version? Solve these classes of problems 19 Suppose some faces of a large wooden cube are painted red and the rest are painted black The cube is then cut into unit cubes The number of unit cubes with some red paint is found to be exactly 200 larger than the number of cubes with some black paint How many cubes have no paint at all? 20 Suppose some faces of an n n n wooden cube are painted red and the rest are painted black The cube is then cut into unit cubes Let R denote the number of cubes with some red paint and B the number 4

5 of cubes with some black paint What is the least value of n for which B + R is a multiple of 100? Find the next five values of n for which B + R is a multiple of 100 In each case decide how the faces of the big cube are painted 21 (2004 Purple Comet) A cubic block with dimensions n n n is made up of a collection of n 3 unit cubes What is the smallest value of n so that if the outer two layers of unit cubes are removed from the block, more than half the original unit cubes will still remain? 22 The cube shown below is built from 125 unit cubes The dots on the surface show the places where the big cube is drilled through When all these drilled cubes have been removed, how many remain? Using perspectives 23 This problem is about using cubes to build polyhedra that resemble buildings Students can practice spacial visualization and use their imagination Here s a sample problem Find all possible cubical buildings that have a base, and front projection and a right side projection that is Solution There are seven solutions We depict them using the base diagram where the number in each square represents the number of 5

6 cubes on top of that square For example 2 2 is the solution that 2 2 uses the maximum number of cubes, whereas 1 2 uses the minimum 2 1 number of cubes How many solutions are there altogether? Answer: 7 Two solutions can be built with 6 cubes, four with 7 cubes and one with 8 cubes For each problem below, build a model that is consistent with the given perspectives (a) Is the number of cubes required determined by the three perspectives? Top view Front view Right view (b) Is the number of cubes required determined by the three perspectives? Top view Front view Right view (c) Is the number of cubes required determined by the three perspectives? What are the maximum number and minimum number of cubes needed to build the model? Build a model for it Top view Front view Right view (d) The top and front projections are given Build a possible right view How many possible right views are there? Top view Front view 6

7 (e) Use exactly 20 cubes to make a model from the building plans below Record the base plan for your building What are the maximum and minimum numbers of cubes that could be used to build the structure Top view Front view Right view 24 A square is decomposed into exactly 75 squares of various (integer) sizes How many 3 3 squares are in this decomposition? 25 A square is decomposed into exactly n squares of various (integer) sizes For which values of n is this possible? 26 Kyle will use four identical unit cubes to create a solid, called a 4- polycube Each cube must be glued to at least one other cube Two cubes may only be glued together in such a way that a face of one cube exactly covers a face of the other cube How many distinct solids could Kyle create? Two solids are considered the same if they can be oriented to they are identical 27 Is it possible to tile a rectangle with squares all of which are different? 28 Farmer Brown s plot is 225 feet long by 90 feet wide He is subdividing it into congruent integer sided rectangular plots (a) How many options are there? (b) For how many of these options are the rectangles actually squares? 29 A positive integer n bigger than 1 can be split into two positive integer summands a and b, usually in several ways Notice that 1 cannot be split The value of the split is a number V (a, b) Starting with an integer n greater than 1, we can successively split n and its offspring to eventually arrive at all 1s Since each splitting adds 1 to the number of unsplit numbers, the number of splits must be exactly n 1 For example we could begin with the number 20 We would find that the 7

8 splitting process is repeated 19 times until the only unsplit numbers are 1 s Specifically, suppose V (a, b) = ab Let n = 20 What is the greatest possible sum of the values of the 19 splits? What is the least possible sum of the values of the 19 splits? 30 Suppose all six faces of an a b c, a b c block of unit cubes are painted Let F abc be the quotient M/abc where M is the number of cubes with some paint For which rational numbers r (0, 1) does there exist a triplet satisfying r = F abc? 31 A large wooden cube is painted on some of its six faces and then cut into small identical cubes The number of small cubes found to have some paint in 15 How many small cubes have no paint at all? Next change the number 15 to 61 and answer the same question 32 What is the greatest number of blocks that can fit in a box? 33 A cube is built from 125 unit cubes Some cubes are removed leaving a smaller cube How many ways can this be done? (For example, there are 8 ways to remove a particular set of 61 unit cubes to leave a subcube 34 A wooden rectangular block, 4 5 6, is painted red and then cut into several 120 unit cubes What is the ratio of the number of cubes with two red faces to the number of cubes with three red faces? 35 Twenty-seven identical white cubes are assembled into a single cube, the outside of which is painted black The cube is then disassembled and the smaller cubes thoroughly shuffled in a bag A blindfolded man (who cannot feel the paint) reassembles the pieces into a cube What is the probability that the outside of this cube is completely black? 36 A cube can be partitioned into 8 cubes in an obvious way A cube can also be partitioned into 27 cubes and into 17 cubes What is the largest integer N such that a cube cannot be partitioned into N cubes? 37 In this problem you re given a cube with an integer value assigned to each face Our first hurdle is to figure out how to translate back and 8

9 forth from a physical cube to a flat piece of paper The mathematical object we deal with is called a net A net is a planar representation of a three-dimensional polyhedron A net for a cube is shown below Think of a cube positioned so that it has a top, a front, left and right sides, a back and a bottom(on the desk) top left front right back bottom 1 For example, it could be just as the integers 1 through 6 6 appear on a standard die Next assign each vertex the product of the numbers in the faces that vertex belongs to For example, the vertex in the top right corner of the face with 3 would have an assigned value of = 15 (a) Let T denote the sum of the eight values of the vertices for the die above Compute T and explain why you get this unusual number While you re thinking about this problem, imagine what happens to the T value is you interchange the numbers on the top and bottom of your cube leaving the other four numbers in place (b) What is the largest possible value of T that can be obtained? Can you prove it? (c) Next we explore the question How many different values of T are obtainable? Assuming that the faces must be labeled with all six digits 1 through 6, how many different values of T can be obtained? (d) Again each face is assigned a positive integer Do not assume here that the integers assigned at 1 though 6 Also, we can assign the same integer to more than one face Suppose the sum T is 70 Can you determine the sum of the six integer faces? (e) Use the nets provided to find as many different T values as you can 9

10 T = T = T = T = T = T = T = T = T = T = T = T = 10

11 T = T = T = 38 Consider next the octahedral net shown below Again, assign to each vertex of the polyhedron (here an octagon) the product of the faces adjacent to that vertex, and again let T denote the sum of the vertex values Find T for the values assigned in the figure What is the largest value T can have if the numbers 1 to 8 are distributed among the faces of the octahedron? 39 A polyhedron has faces that are triangles or squares No two squares share an edge and no two triangles share an edge What is the ratio of the number of triangular faces to the number of square faces? 40 An a b c, 2 a b c rectangular block is built from abc unit cubes From one corner you can see faces of three different sizes Suppose you can see exactly 36 of the abc cubes What is a 2 + b 2 + c 2? 41 Corners are sliced off a unit cube so that the six faces each become regular octagons What is the total volume of the removed tetrahedra? 42 Say you re given the following challenge: create a set of five rectangles that have sides of length 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 units You can 11

12 combine sides in a variety of ways: for example, you could create a set of rectangles with dimensions 1 3, 2 4, 5 7, 6 8 and 9 10 (a) How many different sets of five rectangles are possible? (b) What are the maximum and minimum values for the total areas of the five rectangles? (c) What other values for the total areas of the five rectangles are possible? (d) Which sets of rectangles may be assembled to form a square? 43 The Soma cube was invented by Piet Hein, in the 1930 s It has seven pieces, which are all the ways 3 or 4 cubes can be joined face-to-face, so that the resulting shape is NOT rectangular Four of these pieces are flat and three require three dimensions The flat ones are shown below Z shape L shape T shape V shape The other three are called Y, left hand and right hand (a) Try to put the 7 pieces into the cube (But don t try too long, look at the suggestions below for some hints (b) Look at the pieces i How many cubes are in each piece? ii How many different pieces could you make with three cubes? With four cubes? With five cubes? (Including the rectangular ones) iii Which pieces are symmetric? Which have different mirror images? (c) Here s a sub-problem that might help you as you think about the Soma cube i Look at a checkerboard with two missing corners that are diagonally opposite each other ii Can you tile the checkerboard with 31 dominos? iii Imagine the dominos are colored so that each domino has a black square and a white square 12

13 iv How many black squares are on the checkerboard? How many white squares? How many on each domino? How many are they total in all the 31 dominoes? Can you see why that makes this problem impossible? (d) Look at Soma cube that you are trying to pack How many corners (vertices) does it have? (e) Look at the pieces: how many corners can each one fill? What s the maximum number for each piece, and the minimum number? (f) Note that only ONE piece can fill less than it s maximum corners (g) Note that the T piece must either fill no corners, or 2 corners Can it fill no corners in the solution? Try this out Where must it go? (h) Now imagine the big cube is colored like a 3-dimensional checkerboard Are all the corners the same color? Edges? Facecenters? Center of the whole cube? Let all the corners be colored black How many other cubes are black? (i) Look at each piece If you checkerboard the pieces with two colors, which pieces have an equal number of colored sub-cubes, which have different numbers? (j) If the T piece has to go on an edge, where can the Y piece go? (The Y piece is the one that looks like a corner, and is one of the ones that require three dimensions) (k) The V piece also has limited places it can go, but it is easier to just put the rest of the pieces in (after the T and Y), and save the v for last (l) Can you make a table to show where each piece could go in the cube? (m) Can you guess how many solutions there are in total for the Soma cube? Can you figure an upper-bound for the number of solutions (n) There are MANY ways to dissect a cube into pieces like the Soma cube, giving very many possible puzzles Some are more difficult then Soma, and some are easier Some have a single solution, while others (like the Soma) that have many solutions 13

14 (o) The last question is much harder than all the rest Note that we have built all but two of the four cell (quad) pieces The 2 2 square can be used to replace one of the other quads pieces But which ones? Build a square Notice that the square must have two black and two white cubes no matter how it fits in a Soma cube Does this mean that it cannot replace the Y or the T? Which of the quads can the square replace? 44 A model of a square pyramid is built from unit cubes in such a way that each square layer has one fewer cubes on the side as the layer below it So, for example, if the bottom layer is 3 3 then the middle layer would be 2 2 and the top layer would have just one cube For this set of problems, suppore the base is (a) What is the volume of the model? In other words, how many cubes are needed to build the model? (b) When the model is placed on a table so that the bottom is not visible, what is the surface area of the visible part 45 Three cubes with volumes 1, 8, and 27 are glued together to obtain a solid polyhedron with minimal surface area What is that surface area? 46 Four cubes with volumes 0125, 1, 8, and 27 are glued together to obtain a solid polyhedron with minimal surface area What is that surface area? 47 An a b c, a b c blocks has the same volume numerically as surface area Find all triplets (a, b, c) for which this occurs 48 Color the surface of a cube of dimension red, and then cut the cube into unit cubes Remove all the unit cubes with no red faces Use the remaining cubes to build a cuboid (a rectangular brick), keeping the outer surface of the cuboid red What is the maximum possible volume of the cuboid? 14